Projecting onto a particular Zernike

[jaredsagendorf]

Hello, suppose I have a function (or equivalently ).

What I would like to do is compute the integral (numerically):
for a given n,m

I'm not sure if this is possible with your code, but if so any advice to get started? Thanks!

[Jufevic]

Hello, you could try this:

from zernike import RZern
import numpy as np


def nm2noll(n, m):
    """Convert indices `(n, m)` to the Noll's index `k`."""
    k = n * (n + 1) // 2 + abs(m)
    if (m <= 0 and n % 4 in (0, 1)) or (m >= 0 and n % 4 in (2, 3)):
        k += 1
    return k


def zernike_moment(f, n, m):
    """Compute the projection of the function `f` over `Z_{nm}`.

    :math: `\iint f(x, y) Z_{nm}(x, y) dx dy`
    """
    k = nm2noll(n, m) - 1
    pupil = RZern(n)
    K, L = f.shape
    ddy = np.linspace(-1.0, 1.0, K)
    ddx = np.linspace(-1.0, 1.0, L)
    xv, yv = np.meshgrid(ddx, ddy)
    pupil.make_cart_grid(xv, yv)
    Zk = pupil.matrix(pupil.ZZ[:, k])
    integral = np.nansum(f * Zk)
    return integral / 4

Then, for instance imagine that f is a function that equals to one at every point:

f = np.ones((42, 42))
integral = zernike_moment(f, 1, 1)

Explanation

• This library uses Noll's indexing, that means that if you only know the (n, m) indices of the Zernike polynomial, your first need to convert it as I did in the nm2noll function.
• Then I initialized the Zernike polynomials up to the radial order n.
• With pupil.make_cart_grid, I calculated the Zernike coefficients on a cartesian grid defined by the shape of f.
• Finally to compute the integral, I summed the element-wise multiplication or f and Zk. This gives us an integral. I used np.nansum to ignore nan values in the sum.

Still I'm not certain that the normalization at the end is correct, maybe the result should only be integral. But I hope this will help you getting started!